introduction to gear trainsintroduction to gear...
TRANSCRIPT
Introduction to Gear TrainsIntroduction to Gear Trains
Most of the pictures and animations are from Norton “Design of Machinery”
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Most of the pictures and animations are from Norton, Design of Machinery McGrawHill, 2004
GEAR TRAINS
PinionPinion
RackRack & Pinion W G
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Rack & Pinion Worm Gear
quieter than spur gearp g
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Very high gear ratio is possible in small package. Allow one directional drive: worm →worm wheel
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Pitch circlevP
T 2
d2 3P
d2 3 d2
3P
d
vP
T2T3
V V r r= = =ω ω
3P
T3V V r rP P3 2 13 3 12 2= = =ω ω
13 13 2 223
n r dRn r d
ωω
= = = − = −2
12 12 3 3n r dω
Two gears can only mesh if they have the same “circular pitch”, “diametral pitch” or “module”
13 13 2 223
12 12 3 3
n r dRn r d
ωω
= = = + = +
Circular pitch, cp: 2πr = cpT (T= no of tooth on the circumference)cp= the distance between a point on one gear tooth and the same point on the next gear tooth (cp is measured in inches).Diametral Pitch PD= T/D
Module, m: πD = m T (m is measured in mm)n r d T13 13 2 2 2ω
The + sign is used here to take into account the direction of rotation.
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Hence: Rnn
rr
dd
TT23
13
12
13
12
2
3
2
3
2
3
= = = ± = ± = ±ωω
Simple Gear Trains⎛ ⎞⎛ ⎞
R R RTT
TT
TT24 23 34
2
3
3
4
2
4
= = −⎛⎝⎜
⎞⎠⎟ −⎛⎝⎜
⎞⎠⎟ =
Gear 3 is “idler” Used to change theGear 3 is idler . Used to change the direction of rotation or to connect two shafts at a distance.
Simple “Compound”Gear Train
n n n n T T T T/ /
( )Rn n n nn n n n
R R R RT T T TT T T T26
16 15 14 13
15 14 13 1223 34 45 56
4 2 3 4 5
3 4 5 6
1= = = −
In general:
( )Rnnij
j
i
k= = −1
1
1 Product of driving gear tooth numbersProduct of driven gear tooth numbers
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k = number of external gear meshes.
Reverted GearUsed in automotiveUsed in automotive transmission:
- compact, save space
Revert = go back to a previous stateRevert = go back to a previous state
Speed change gear box
0-4-5-6-10-12
2 22 23 11( 1) 0.32334 46 34
output
input
ωω
•= − = =
•
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Important: Gears are numbered. Not the links!!
Input Shaft1
Intermidiate Shaft10.80640.6470.514
1-92-84-53-7
Intermidiate Shaft0.514
7-116-10
Output Shaft
1.156
0.932
0.748
0.594
0.5000.4030.3240.257
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Automotive Transmission Low gear: Gear 3 meshes with gear 6, power flows 1-4-6-3
14 18 0.30131 27
out
in
ωω
= • =
Second Gear: gear 2 meshes with gear 5, power flows 1-4-5-2
14 25
High Gear: gear 2 is shifted so that
14 25 0.56431 20
out
in
ωω
= • =
High Gear: gear 2 is shifted so that the clutch teeth on the end of gear 2 mesh with the clutch on gear 1 (direct drive)drive)
Reverse gear: gear 3 is shifted to
1out
in
ωω
=http://auto.howstuffworks.com/transmission.htm
Reverse gear: gear 3 is shifted to mesh with gear 8, power flows 1-4-7-8-3.
14 14 0 234outω
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0.23431 27
out
inω= − • =
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Dual Clutch Transmission
From: How stuffs work: http://auto howstuffworks com/dual clutch transmission1 htm
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From: How stuffs work: http://auto.howstuffworks.com/dual-clutch-transmission1.htm
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PLANETARY GEAR TRAINSThe axis of one of the gear (planet) is not fixed
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Planetary Gear Trains
APlanet
i i ij
iA
k k k kArmArm
jj
jA0
Sun Gear
AA0
0
Sun Gear
Ring Gear
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VP/A P
VA
VP =VP
AVA
ω
Aω1i j
1i
i
AP/AV
PP=VVP
ωω 1k 1k
Aj
i
j
ii
ωA A
1j00j
∓ω 1j
∓
VPi = VPj = VA + VP/A
1
( )Pj Pi j jv v r
v r r
ω
ω
= =
= ∓(- if external, + if internal mesh)
1
/ 1
1 1 1
( )
( )
A k j i
P A i i
j j k j i i i
v r r
v rr r r r
ω
ωω ω ω
=
= ±= ±
∓
∓ d T
± =−ri j kω ω1 1
1 1 1( )j j k j i i ir r r rω ω ω±∓± = ± = ± =
rr
dd
TT
Ri
j
i
j
i
jij Gear Ratio
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−rj i kω ω1 1 Gear Ratio is not equal to the speed ratio
ωarm=ω14=40s-1 (CW)
ωsun=ω12=?
80 20ω ω12 14
11 14
80 20 220 40
ω ωω ω
−= − = −
−i
0ω11=0
12 14 14 142( ) 3ω ω ω ω= − − =
ωsun = ω12 =120 s-1 (CW)
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T'
k
T
p
j
T
j Rijki
ki
j k
i k
= =−
−ωω
ω ω
ω ω1 1
1 1
k ip
Tj R pjkj
kp
j k
p k
= =−
−
ω
ω
ω ω
ω ω1 1
1 1
1R
Rpj
jp=
RRRip
ij
pj
p k
i k
= =−
−
ω ω
ω ω1 1
1 1
Tp ( )R R RT TT Tip ij jp
k j i
j p
= = −1 /
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90T2-3 and 2-4 are simple gear trains
30T
5 6
n14= - 3624
n12 n14= - 32
n12
20T 324T
5
7
70T20T
6
n13= - 4020
n12 n13= -2 n12
3
n12=100rpm 30T
4
2
7Now consider links 3,4,5,6 and 7.
Link 5 is the planet, link 3 is the arm
40T 36Tn17-n13
n14-n13
30*30*20
90*20*70= - = - 1
7
n17= 87
n14n13- 1
7 297 7
n17= -8*2 n12n - -3
n17= - 2914
n12
207 1418
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n17 7
n12n127*2 n17= - 207.14 rpm
91T
3 5
91T 92T 92T
PlanetPlanet
n12= 2000 rpm
n16=?n16 ?
First planet (arm red- link2)n14-n12 90*92
2 64Arm Arm
14 12
n11-n12
= 90 92
91*91
Since n 0 :
90T 90T91T 91T
Since n11=0 :
n14= n12-90*92
91*91 90T 90T91T 91T91*91 n12
n14= 8281-82808281
n12
Second planet (arm blue- link4)
n16-n14 = 90*92 n16= n14-90*92
8281
n14= 18281
n12
n11-n14
=91*91
16 1491*91
n = 8281-8280 n14
n14
n16= 1 n148281 n16=8281
148281
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n16=1 n12 n16= 1 n n16= 0.0000292 rpm
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n1682812
n12 1668,574,961
n12
Example: Model T Ford gearboxGearbox : Integral with the engine. 2: On g gFoot operated 2 speed and reverse epicyclic transmission foot-brake, 1908 for 19 yrs9 million were made!
P1P2
2: On
http://www.t-ford.co.uk/car.htm
I/PO/P
S1S2
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Low gear for the model T Ford
nout-nin
ns1-nin
= 21*2733*27
= 7/11
ωin =0.3636 ninnout=(1-7/11) nin =4/11nin
ωout
Fixed
Reverse gear for the model T Ford
nout-nin
ns1-nin
= 30*2724*27
= 5/4
= - 0.25 ninnout=(1-5/4) nin =-1/4nin
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Fixed
GEAR TRAINS WITH BEVEL GEARST3r3r3
+
θ β
P
r2
O
θ
α
β
T2
( )ω13 2 2 2r T r OP/( )( )ω
13
12
2
3
2
3
2
3
= = =r T r OP/
= =siniαβ
R 23sinβ
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P1
80T
60T
4+
+
3
O
56T3 +
2+
24T4
P2
2Input Output 2
541
P2
18T
20T
Simple compound gear train ( f ll fi d )
+
20T35T
76T
+
Planetary Gear Train(axes of all gears are fixed axes)
R 2313
12
2080
= =ωω
and
R n nn n12
12 15
11 15
76 5656 20
195
=−−
= − = −**
5R 34
14
13
1860
= =ωω
R 14 20 18 3= = = =ω * Product of driving gear tooth number
or
n n15 125
24=Since n11=0:
Rn n
1414 15 76 24
56 35228245
=−
= + = +**R 24
12 80 60 40= = = =ω * Product of driven gear tooth number
075.0403R
12
1424 +=+=
ωω
=
n n1411 15 56 35 245− *
Since n11=0:
1 228 17 17 5⎛⎜
⎞⎟ *n n n n14 15 15 121
245 245 245 24= −⎛⎝⎜
⎞⎠⎟
= = *
Nn
2414 0 0145= = .
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n2412
28T 56 3
92TMotion from 2 to 3 and 2 to 4 are simple gear trains (axes fixed):
90T
30T
4n14= 5020
n12
50 50T20T
INPUT
n13= 5020
n12
Links 3 and 4 rotate in different directions
2
20T
INPUT
n13= - n14= 2.5 n12
Considering links 3,4,5 and 6; link 6 is the planet and link 5 is the arm (output)
O
planet and link 5 is the arm (output)
n13-n15
n -n90*28
30*92= +
OUTPUTn14-n15 30*92
n13-n15 = 2123
(n14-n15)232
23n15 = n13 - 21
23 n14
2 5n15 = 55 n12
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2
23n15 = 44
23n12
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